A heuristic inspection method to find causes of system abnormality based on dynamic uncertain causality graph

ABSTRACT

Provided is a method for ordering X-type variables having states to be measured in a dynamic uncertain causality graph (DUCG). The method comprises: step 1: determining, on the basis of a DUCG simplified using E(y) as a condition, a measurable X-type variable having a state to be measured, an index set thereof being S x(y), where y is a time series; step 2: if there is only one element in S x(y), ending ordering; step 3: calculating ranking importance Ii(y) for Xi(i∈S x(y)); step 4: ordering ranking Xi(i∈S x(y)) according to the ranking importance Ii(y), and performing state measurement on an X-type variable of i∈S x(y) by referring to the ranking; and step 5: adding 1 to y, and repeating steps 1-5 until there is no X-type variable to be measured. The technical solution of the invention can quickly diagnose a cause of an object system abnormality at minimum cost, and effectively returns the object system to normal.

FIELD OF THE INVENTION

This invention is about an AI technology to process uncertain causality type information represented by Dynamic Uncertain Causality Graph (DUCG). Based on the technical scheme proposed in this invention, through computation with computer, one can rank the variables to be detected optimally, detect the ranked variables sequentially or group by group to find out their states, to diagnose causes of system abnormality as earlier and less cost as possible, and then enable to take effective actions to make system normal.

BACKGROUND OF THE INVENTION

There exist many cause events that may lead to abnormalities of industrial systems, social systems and biological systems, such as short circuit of coils, pump fails to run, failure of components, malfunction of sub-systems, blocking of pathway, entry of foreign matter, pollution, infection, damage, decrepitude of tissue or organ, etc. When system abnormality appears, people need to know the real cause event as soon as possible. Denote B_(n) or BX_(n) as these cause event variables indexed by n and B_(nk) or BX_(nk) as the event that B_(n) or BX_(n) is in state k. The difference between B_(n) and BX_(n) is that B_(n) represents root-cause variable without input while BX has inputs reflecting the influence of other factors. Usually, k=0 means that B_(n) or BX_(n) is in normal state; k=1, 2, 3 . . . means that B_(n) or BX_(n) is in abnormal state. Most of the states of B_(n) and BX cannot be or are hard to be detected directly. Furthermore, there are a large number of factors that have causal relationships to B_(n) or BX_(n), such as temperature, pressure, flow, velocity, frequency, various chemical or physical test results, investigation results, imaging examination results, acoustic examination results, and so on. Some factors may increase or decrease the occurrence probability of B_(nk) (k≠0), e.g. region, time, environment, season, religion, age, sex, skin color, career, blood relation, hobbies, personality, living conditions, working conditions, etc. The affected B_(nk) is BX_(nk). All these factors can be represented by event variable X_(i), while X_(y) represents state j of X_(i). X_(i) and X_(ij) also represent cause or consequence of other variables/events. Usually, j=0 means that X is in normal state; j=1, 2, 3 . . . means that X is in abnormal state. By detecting the states of X_(i), people can diagnose the root cause (B_(nk) or BX_(nk) (k≠0)) of system abnormality, so that be able to take effective actions in time to get the system back to normal or reduce damage. Dynamic Uncertain Causality Graph (DUCG) is an AI scheme to represent the uncertain causal relationships among event variables graphically, and perform diagnoses based on the constructed graph and observed evidence E composed of known states of X-type variables X_(ij). For example, E=X_(1,2)X_(2,3)X_(3,1)X_(4,0)X_(5,0), where comma separates the variable index (the first) and state index (the second). In general, the more the state known X-type variables in E, the more accurate the diagnosis can be. Yet some state known X-type variables contribute little or none to the diagnosis, some contribute a lot. In practice, there exist cost to detect the states of X-type variables. Therefore, one may have to choose some to detect, or detect some earlier and some later. The problem to be solved in this invention is as follows:

Based on

-   -   (1) the constructed DUCG,     -   (2) evidence E(y) observed by the time indexed with y (y=0, 1,         2, . . . ),     -   (3) possible cause events included in hypothesis set S_(H)(y)         diagnosed conditional on E(y),         which state-unknown and detectable X-type variable should be         detected to update evidence E(y) as E(y+1)=E⁺(y)E(y), so that         more accurate diagnoses S_(H)(y+1) conditional on E(y+1) can be         made as soon and with less cost as possible. Where, the new         detected states of X-type variables are the new evidence denoted         as E⁺(y), the new hypothesis set denoted as S_(H)(y+1) is         diagnosed conditional on E(y+1), and y=0 means the time no         evidence received, that is, E(0) is a complete set denoted as         E(0)=1.

The possible cause event in S_(H)(y) or S_(H)(y+1) is represented as H_(kj), in which H_(k) is one or a set of variables indexed by k, for example, H₁=BX₁, H₂=BX₂B₄, etc., and j indexes the state combination of the set of variables, for example, H_(1,2)=BX_(1,2), H_(2,3)=B_(1,3)X_(4,2), etc.

An example of DUCG is illustrated in FIG. 1. In DUCG, B-type variable or event is drawn as rectangle, X-type variable or event is drawn as circle, BX-type variable or event is drawn as double line circle, G-type variable or event representing the logic relationship is drawn as gate, and D-type variable or event is drawn as pentagon representing the default or unknown cause of X-type variable/event. The {B−, X−, BX−, G−, D−}-type variable/events are also called nodes.

Variable G_(i) represents the logic combinations of input variables. It must have at least two inputs connected with directed arc

. The logic combinations are specified by logic gate specification LGS_(i). For example, in FIG. 1 G₁ is specified by LGS₁: G_(11,1)=B_(3,1)X_(111,1), G_(11,2)=B_(3,1)X_(111,2), G_(1,0)=Remnant State that is defined as all other state combinations.

The textual descriptions of {B−, X−, BX−, D−, G−}-type variables and their states can be given according to their physical meanings. All {B−, X−, BX−, D−, G−}-type variable/event can be direct cause variable/event, and is called parent variable/event, and can be represented by V∈{B, X, BX, D, G} with the same index. For example, V₂=X₂, V_(3,2)=B_(3,2), etc. Direct consequence or child variables/events are usually {X−, BX−}-type variables/events.

Directed arc

from parent to child is used to denote the weighted functional variable F_(n;i) representing the causal relationship between parent variable V_(i) and child variable X_(n) or BX_(n). F_(n;i) is an event matrix. F_(nk;ij)≡(r_(n;i)/r_(n))A_(nk;ij) represents the causal relationship between parent event V_(ij) and child event X_(nk) or BX_(nk). Where r_(n;i)>0 quantifies the uncertain causal relationship intensity between V_(i) and X_(n) or BX_(n), r_(n)≡Σ_(i)r_(n;i), and A_(nk;ij) represents the uncertain causal mechanism that V_(ij) may cause X_(nk) or BX_(nk). The probability of A_(nk;ij) is defined as a_(nk;ij)=Pr{A_(nk;ij)} usually given by domain experts. Define f_(nk;ij)=Pr{F_(nk;ij)}=(r_(nk;ij)/r_(n))a_(nk;ij), in which f_(nk;ij) means the probabilistic contributions from V_(ij) to X_(nk), satisfying

${Pr\left\{ X_{nk} \right\}} = {\sum\limits_{i,j}{f_{{nk};{ij}}\Pr{\left\{ V_{ij} \right\}.}}}$

F_(nk;ij), f_(nk;ij), A_(nk;ij) and a_(nk;ij) are member event in F_(n;i), f_(n;i), A_(n;i) and a_(n;i) respectively. Define v_(ij)=Pr{V_(ij)}, v∈{b, x, bx, d, g}. In general, the lower case letters represent the probabilities of the corresponding upper case letters that represent events or event variables.

The weighted functional event matrix F_(n;i) can also be conditional on a condition event Z_(n;i), which can be drawn as dashed directed arc

. Conditional F_(n;i) is used to represent the conditional causal relationship between its parent event vector V_(i) and its child event vector X_(nk) or BX_(nk). The condition event Z_(n;i) determines whether F_(n;i) holds or not (discarded). Taken Z_(n;i)=X_(1,2) as an example, when X_(1,2) is true, F_(n;i) is held, i.e.

becomes

; when X_(1,2) is false, F_(n;i) is not held, i.e.

is discarded.

For simplicity, the letter of a variable in each graphical node symbol can be ignored and only the indices are inside as shown in FIG. 2, in which the first number is the index of the variable and the second number is the state index of the variable. Since D-type variable has only one state, there is only one index inside the symbol. Non-D-type variables with known state can be indicated by colors, e.g. X_(110,1) is blue in FIG. 2. If there is only one number in the symbol, the state of the variable is unknown.

With the received evidence E(y)=E′(y)E″(y), where E′(y) is composed of state abnormal events and E″(y) is composed of state normal events, the following rules can be used to simplify a DUCG:

Rule 1: If E(y) shows that Z_(n;i) is false, F_(n;i), is eliminated; if E(y) shows that Z_(n;i) is true, the conditional F_(n;i) becomes the ordinary F_(n;i), i.e.

becomes

3. Rule 2: If E(y) shows that V_(ij) (V∈{B, X}) is true while V_(ij) is not a parent event of X_(n) or BX_(n), directed arc F_(n;i) is eliminated.

Rule 3: If E(y) shows that X_(nk) is true while X_(nk) cannot be caused by any state of V_(i) (V∈{B, X, BX, G, D}), directed arc F_(n;i) is eliminated.

Rule 4: If E(y) shows that a state-unknown X-type node does not have any output and E(y) blocks its connection with B-type nodes, eliminate this X-type node. Rule 5: If E(y) shows X_(n0) is true, and X_(n0) has no connection with B-type nodes unless through the nodes in E(y), eliminate X_(n0). Rule 6: If E(y) shows a group of state-unknown nodes that have no connection with E′(y), this group of state-unknown nodes are eliminated. Rule 7: Given E(y), if G_(i) has no output, eliminate G_(i) and its input directed arc

; if G_(i) has no input, eliminate G_(i) and its output directed arcs. Rule 8: Given E(y), if a directed arc has no parent node or child node, eliminate this directed arc. Rule 9: If there is a group of nodes and directed arcs that have no connection with E′(y), this group of nodes and directed arcs are eliminated. Rule 10: If E(y) shows X_(nk) (k≠0) is true while X_(nk) does not have any input, add a virtual parent event D_(n) to X_(nk) as its input with a_(nk;nD)=1 and a_(n′;nD)=0 (k≠k′), where r_(n;D) can be any value. The added virtual D_(n) can be drawn as

Rule 11: Apply Rule 1-Rule 10 in any order, separately or together, and repeatedly.

After applying the above rules, the DUCG is simplified, in which H_(kj)=B_(kj) or H_(kj)=BX_(kj) (j≠0) compose S_(H)(y).

In the simplified DUCG, some state-unknown X-type variables are cause-specific, which means that the state observation of a cause-specific X-type variable can determine whether the corresponding hypothesis event H_(kj) (H_(kj)∈S_(H)(y)) is true.

Technical references for this invention

-   [1] Q. Zhang and Z. Zhang. Method for constructing an intelligent     system processing uncertain causal relationship information. Chinese     patent: CN 200680055266.X, 2010. -   [2] Q. Zhang and Z. Zhang. Method for constructing an intelligent     system processing uncertain causal relationship information. U.S.     Pat. No. 8,255,353 B2, 2012. -   [3] Q. Zhang and C. Dong. Method for constructing cubic DUCG for     dynamic fault diagnosis. Chinese patent: CN 2013107185964, 2015. -   [4] Q. Zhang. “Dynamic uncertain causality graph for knowledge     representation and reasoning: discrete DAG cases”, Journal of     Computer Science and Technology, vol. 27, no. 1, pp. 1-23, 2012. -   [5] Q. Zhang, C. Dong, Y. Cui and Z. Yang. “Dynamic uncertain     causality graph for knowledge representation and probabilistic     reasoning: statistics base, matrix and fault diagnosis”, IEEE Trans.     Neural Networks and Learning Systems, vol. 25, no. 4, pp. 645-663,     2014. -   [6] Q. Zhang. “Dynamic uncertain causality graph for knowledge     representation and probabilistic reasoning: directed cyclic graph     and joint probability distribution”, IEEE Trans. Neural Networks and     Learning Systems, vol. 26, no. 7, pp. 1503-1517, 2015. -   [7] Q. Zhang. “Dynamic uncertain causality graph for knowledge     representation and probabilistic reasoning: continuous variable,     uncertain evidence and failure forecast”, IEEE Trans. Systems, Man     and Cybernetics, vol. 45, no. 7, pp. 990-1003, 2015. -   [8] Q. Zhang and S. Geng. “Dynamic uncertain causality graph applied     to dynamic fault diagnosis of large and complex systems”, IEEE     Trans. Reliability, vol. 64, no. 3, pp 910-927, 2015. -   [9] Q. Zhang and Z. Zhang. “Dynamic uncertain causality graph     applied to dynamic fault diagnoses and predictions with negative     feedbacks”, IEEE Trans. Reliability, vol. 65, no. 2, pp 1030-1044,     2016.

SUMMARY OF THE INVENTION

This invention discloses a technical scheme to rank the state-unknown X-type variables optimally, so that people can optically choose the state-unknown X-type variables to detect (state pending X-type variables) to obtain E⁺(y). Conditional on E(y+1)=E₊(y)E(y), the more accurate S_(H)(y+1) and more accurate Pr{H_(kj)}, H_(kj)∈S_(H)(y+1), can be found as earlier and with less cost as possible, while the danger degrees of H_(kj) are considered.

This invention is a subsequent invention and a further development of granted patents CN 200680055266.X, CN 2013107185964, and U.S. Pat. No. 8,255,353 B2. The technical scheme of this invention comprises:

1. The method to rank the state pending X-type variables with at least one CPU, according to the rank, part of or all of the state pending X-type variables' states which form E⁺(y) are detected sequentially or parallel, in order to find the real cause H_(kj) that is in S_(H)(y+1) and to rank the real H_(kj) as high as possible conditioned on E(y+1)=E⁺(y)E(y), detailed steps include: (1) determine the detectable state pending X-type variables whose index set is denoted as S_(X)(y) based on the simplified DUCG conditioned on E(y); (2) the ranking ends if S_(X)(y) contains only one element; (3) calculate the rank importance I_(i)(y) of X_(i); (4) rank X_(i) (i∈S_(X)(y)) according to I_(i)(y) and detect the states of X_(i) (i∈S_(X)(y)) in reference to the rank; (5) if the ranking is still needing, increase y to y+1, and repeat the above step (1)-(5) until the diagnosis is satisfied or no state pending X-type variable available.

2. The method according to 1, wherein to determine the state pending X-type variables with at least one CPU, the detailed steps include: (1) Collect all possible H_(kj) based on the simplified DUCG conditioned on E(y), these H_(kj) form S_(H)(y); (2) For each H_(k) in S_(H)(y), search for the state pending X-type variable connected to H_(k) without state-known variables blocking them, where the indices of such X-type variables make up S_(X)(y).

3. The method according to 1, wherein to determine the structure importance λ_(i)(y)>0 of X_(i) for calculating I(y) with at least one CPU, characterized in that: for the state pending X-type variables in the above 1(1), count the number of its connected different H_(k) (k∈S_(iK)) in S_(H)(y) determined in the above 2, the number is written as m_(i)(y), calculate λ_(i)(y) based on m_(i)(y) according to a method that features at that the bigger m_(i)(y) is, the smaller λ_(i)(y) is, such method includes but not limits to λ_(i)(y)=1/(m_(i)(y))^(n) (n=1, 2, . . . ).

4. The method according to 1, wherein assign a value to the danger importance ω_(kj)>0 of H_(kj) for calculating I_(i)(y) with at least one CPU, characterized in that:

For each possible cause event H_(kj) in S_(H)(y), score all of the abnormal states according to their degree of concern, the score is called concern importance which is written as ω_(k), 1≥ω_(k)>0. The greater the concern is, the bigger ω_(k) is. The value of ω_(k) can be assigned when constructing DUCG or be assigned according to the concrete situation given H_(k) when S_(H)(y) is known.

5. The method according to 1, wherein to calculate the probability importance ρ_(i)(y) for calculating I(y) with at least one CPU, characterized in that: calculate the average variation of the conditional probabilities of H_(k) in S_(H)(y) between the conditions E(y) and X_(ig)E(y) over all abnormal states of all H_(k) connected with X without state-known variables blocking them, where g∈S_(iG)(y), S_(iG)(y) is the index set of possible states of X_(i), and X_(i) is non-cause-specific state pending X-type variables, based on the simplified DUCG conditioned on E(y), featuring at that the greater the average variation is, the bigger ρ_(i)(y) is, such as but are not limited to:

${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \middle| {Or} \right.$ ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}\left( {Pr\left\{ X_{ig} \middle| {E(y)} \right\}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)} \right)^{2}}}}}}$ Or ${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {S_{kJ}{(y)}}}\sum\limits_{g \in {S_{iG}{(y)}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \middle| {Or} \right.$ ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{(y)}}}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)^{2}}}}}}$ Or ${\rho_{i}(y)} = \left. {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}\omega_{k}}} \middle| {{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}} - {\frac{1}{m_{i}}{\sum\limits_{{k \in {S_{iK}{(y)}}},{j \in {S_{kJ}{(y)}}}}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}}}}} \right|$

E(y) and ω_(k) can be ignored, equivalent to being removed separately or together, that is to let E (y)=the complete set and ω_(k)=1.

6. The method according to 1 for calculating I(y) with at least one CPU, wherein to determine the cost importance, characterized in that: comprehensively assign a cost score for the state pending X_(i) (i∈S_(X)(y)), or calculate the cost score as the sum of the weighted scores assigned for the difficulty of doing the detection (indexed by j=1), waiting time (indexed by j=2), price (indexed by j=3) and damage to target system (indexed by j=4) respectively, the weights σ_(ij) can be given when constructing DUCG or be given in an individual application, the bigger the cost score is, the smaller the cost importance β_(i) (1≥β_(i)>0) is, which including but not limited to: when the highest cost score is 100, β_(i)= 1/100=0.01, when the lowest cost score is 1, β_(i)=1/1=1, and the remnant values are between the highest and the lowest case, can be assigned when constructing DUCG or be given online for the state pending X-type variables included in the simplified DUCG conditioned on E(y).

7. The method according to 5 to determine the probability importance ρ_(i)(y) with at least one CPU, characterized in that: based on the simplified DUCG conditioned on E(y), for each cause-specific state pending X-type variable connected to H_(kj) (H_(kj) ∈S_(k)(y)), λ_(i)(y) and ρ_(i)(y) are not calculated according to the above 3 and 5, which means S_(Xs)(y) is obtained by subtracting S_(s)(y) from S_(X)(y), but are calculated by selecting the maximum of i∈S_(s)(y), which includes but is not limited to λ_(i)(y)≥1 and

${{\rho_{i}(y)} \geq {\max\limits_{l \in {S_{Xs}{(y)}}}\left\{ {\rho_{l}(y)} \right\}}},$

that means X_(i) in the above 5 is limited to i∈S_(Xs)(y).

8. The method according to 1 to comprehensively calculate I(y) with at least one CPU, characterized in that: the bigger λ_(i)(y) or ρ_(i)(y) or β_(i) in the above 2-7 is, the bigger I_(i)(y) is, the detailed calculation formulas include but are not limited to:

${I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}$ Or I_(i)(y) = λ_(i)(y)β_(i)ρ_(i)(y) Or ${I_{i}(y)} = \frac{{\lambda_{i}(y)}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}{\rho_{i}(y)}}}$ Or I_(i)(y) = λ_(i)(y)ρ_(i)(y) Or ${I_{i}(y)} = \frac{\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{\beta_{i}{\rho_{i}(y)}}}$ Or I_(i)(y) = β_(i)ρ_(i)(y) Or ${I_{i}(y)} = \frac{\rho_{i}(y)}{\sum_{i \in {S_{X}{(y)}}}{\rho_{i}(y)}}$ Or I_(i)(y) = ρ_(i)(y) Or I_(i)(y) = w₁λ_(i)(y) + w₂ρ_(i)(y) + w₃β_(i) Or ${I_{i}(y)} = \frac{{w_{1}{\lambda_{i}(y)}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}{{\sum_{i \in {S_{X}{(y)}}}{w_{1}{\lambda_{i}(y)}}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}$

where w₁, w₂, and w₃ are three weights, and w_(i)≥0 (i=1, 2, 3), and w_(i)=0 means this item is not considered, wherein the value of w₁, w₂, and w₃ can be assigned when constructing DUCG or be assigned according to the individual application situation.

9. The method according to 1 to rank the state pending X-type variables X_(i) (i∈S_(X)(y)) according to I_(i)(y) with at least one CPU, characterized in that: in the ranking, when the top-ranking X-type variable is the only ancestor or descendant variable of the ranking lower X-type variable, the ranking lower X-type variable is eliminated from the current rank, also the X_(i) whose I_(i)(y)=0 is eliminated from the rank.

10. The method according to 1 to determine whether the ranking ends or not with at least one CPU, characterized in that: the ranking ends if there is only one hypothesis event in S_(H)(y), or if all the state pending X-type variables are state-known, or there is only one element in S_(X)(y).

The chart of the above steps is as shown in FIG. 3.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1: Illustration of DUCG.

FIG. 2: Illustration of a simple representation of DUCG.

FIG. 3: Step chart of this invention.

FIG. 4: The original DUCG of the examples.

FIG. 5: The results of dividing and simplifying FIG. 4 based on BX₁ when y=0.

FIG. 6: The results of dividing and simplifying FIG. 4 based on BX₂ when y=0.

FIG. 7: The results of dividing and simplifying FIG. 4 based on BX₃ when y=0.

FIG. 8: The DUCG graph of example 1 after detecting the X-type variables ranking first 5.

FIG. 9: The simplified DUCG when y=1.

FIG. 10: The results of dividing and simplifying FIG. 9 based on BX₁ when y=1.

FIG. 11: The results of dividing and simplifying FIG. 9 based on BX₂ when y=2.

FIG. 12: The DUCG graph after detecting the states of X₈ and X₁₀.

FIG. 13: The simplified DUCG graph given E(2).

FIG. 14: The results of dividing and simplifying FIG. 13 based on BX₁.

FIG. 15: The results of dividing and simplifying FIG. 13 based on BX₂.

EXAMPLES TO IMPLEMENT THIS INVENTION

Suppose FIG. 4 is the original DUCG graph whose parameters are as follows:

${b_{1} = \begin{pmatrix}  - & 0.08 \end{pmatrix}^{T}};{b_{2} = \begin{pmatrix}  - & 0.01 & 0.02 \end{pmatrix}^{T}};{b_{3} = \begin{pmatrix}  - & 0.05 \end{pmatrix}^{T}};{a_{1;1} = \begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}};$ ${a_{2;2} = \begin{pmatrix}  - & - & - \\  - & 1 & - \\  - & - & 1 \end{pmatrix}};{a_{3;3} = \begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}};{a_{12;D} = \begin{pmatrix}  - \\ 0.1 \end{pmatrix}};{a_{13;D} = \begin{pmatrix}  - \\ 0.01 \end{pmatrix}};{a_{14;D} = \begin{pmatrix}  - \\ 0.1 \end{pmatrix}};$ ${a_{1;12} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{1;13} = \begin{pmatrix}  - & - \\  - & 0.8 \end{pmatrix}};{a_{3;14} = \begin{pmatrix}  - & - \\  - & 0.6 \end{pmatrix}};{a_{4;1} = \begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}};$ $\left( {{a_{4;2} = \left( {\begin{matrix}  - \\  -  \end{matrix}\begin{matrix}  - & - \\ 0.5 & 0.5 \end{matrix}} \right)};{a_{5;1} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{5;2} = \left( {\begin{matrix}  - \\  -  \end{matrix}\begin{matrix}  - & - \\ 0.4 & 0.8 \end{matrix}} \right)};{a_{5;3} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{6;2} = \begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}};{a_{6;3} = \begin{pmatrix}  - & - \\  - & 0.2 \\  - & 0.8 \end{pmatrix}};{a_{7;4} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{8;4} = \begin{pmatrix}  - & - \\  - & 0.8 \end{pmatrix}};{a_{8;5} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{8;6} = \begin{pmatrix}  - & - & - \\  - & 0.2 & 0.8 \end{pmatrix}};{a_{9;4} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{10;5} = \begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}};{a_{11;6} = \begin{pmatrix}  - & - & - \\  - & 0.3 & 0.8 \end{pmatrix}};{a_{15;3} = \begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}};{a_{16;3} = \begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}};} \right.$

r_(n;i)=1 are assumed. Among all the X-type variables, X₁₄ is a non-detectable variable; other X-type variables are all detectable. X₁₅ and X₁₆ are cause-specific variables of BX_(3,1), that is when X_(15,1) or X_(16,1) is true, BX_(3,1) must be true. Since {B_(1,1), B_(2,1), B_(2,2), B_(3,1)} is equivalent to {BX_(1,1), BX_(2,1), BX_(2,2), BX_(3,1)}, the B-type variables are not treated as the diagnosis object. In other words, H-type events are composed of only BX-type events.

The parameter of the directed arc between X_(n) and D_(n) can be briefly written as a_(n;D). Our task is to detect the states of the state pending X-type variables as few and with less cost as possible, in order to minimize the size of the possible set of cause events S_(H)(y) and to maximize the probability of real cause event.

Example 1: y=0 (with No Evidence)

According to 2, when there is no evidence (y=0), based on FIG. 4, S_(H)(0)={H_(1,1), H_(2,1), H_(2,2), H_(3,1)}={BX_(1,1), BX_(2,1), BX_(2,2), BX_(3,1)}, and the state pending X-type variables are {X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₅, X₁₆}. All these states are unknown and single-connect to H_(k) in S_(H)(0) without state-known blocking variables, so that S_(X)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16}. Since X₁₅ and X₁₆ are cause-specific variables of BX_(3,1), they are eliminated from S_(X)(0), that means S_(Xs)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. Accordingly, S_(4G)(0)=S_(5G)(0)=S_(7G)(0)=S_(8G)(0)=S_(9G)(0)=S_(10G)(0)=S_(11G)(0)=S_(12G)(0)=S_(13G)(0)={1} and S_(6G)(0)={1, 2}.

Based on FIG. 4, according to the DUCG algorithms in Ref [8], since the intersection of different H_(kj) is null set, which means different H_(kj) cannot occur simultaneously, FIG. 4 can be divided according to H_(k) and can be simplified according to simplification rules, the results are shown in FIG. 5-FIG. 7.

Since there is no evidence, E(0)=1 (complete set), the probability and weighting factor of each sub-graph are ζ_(i)(y)=ζ_(i)(0)=Pr{E(0)}=1 (i∈{1,2,3}) and

${\xi_{i}(y)} = {{\xi_{i}(0)} = {{{\zeta_{i}(0)}/{\sum\limits_{i}{\zeta_{i}(0)}}} = {{\frac{1}{3}{\xi_{i}(y)}} = {{\xi_{i}(0)} = {{{\zeta_{i}(0)}/{\sum\limits_{i}{\zeta_{i}(0)}}} = {\frac{1}{3}.}}}}}}$

Based on FIG. 5-FIG. 7, according to algorithms in the references, the state probabilities h_(kj) ^(s)(y)=h_(kj) ^(s)(0) of BX_(1,1), BX_(2,1), BX_(2,2) and BX_(3,1) are:

$\begin{matrix} {{h_{1,1}^{s}(0)} = {{\xi_{1}(0)}\Pr\left\{ {{BX}_{1,1}❘{E(0)}} \right\}}} \\ {= {{\xi_{1}(0)}\Pr\left\{ {BX}_{1,1} \right\}}} \\ {= {{\xi_{1}(0)}\Pr\left\{ {{\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{1,{1;12}}A_{12;D}} + {\frac{r_{1;13}}{r_{1}}A_{1,{1;13}}A_{13;D}}} \right\}}} \\ {= {{\frac{1}{3} \times \frac{1}{3}a_{1,{1;1}}b_{1}} + {\frac{1}{3}a_{1,{1;12}}a_{12;D}} + {\frac{1}{3}a_{1,{1;13}}a_{13;D}}}} \\ {= {{\frac{1}{3} \times \frac{1}{3}\begin{pmatrix}  - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.08 \end{pmatrix}} + {\frac{1}{3}\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - \\ 0.1 \end{pmatrix}} + {\frac{1}{3}\begin{pmatrix}  - & 0.8 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \end{pmatrix}}}} \\ {= 0.01756} \end{matrix}$ $\begin{matrix} {{h_{2,1}^{s}(0)} = {{\xi_{2}(0)}\Pr\left\{ {BX}_{2,1} \right\}}} \\ {= {{\xi_{2}(0)}\Pr\left\{ {A_{2,{1;2}}B_{2}} \right\}}} \\ {= {{\xi_{2}(0)}a_{2,{1;2}}b_{2}}} \\ {= {\frac{1}{3}\begin{pmatrix}  - & 1 & -  \end{pmatrix}\begin{pmatrix}  - & 0.01 & 0.02 \end{pmatrix}^{T}}} \\ {= 0.003333} \end{matrix}$ $\begin{matrix} {{h_{2,2}^{s}(0)} = {{\xi_{2}(0)}\Pr\left\{ {BX}_{2,2} \right\}}} \\ {= {{\xi_{2}(0)}\Pr\left\{ {A_{2,{2;2}}B_{2}} \right\}}} \\ {= {{\xi_{2}(0)}a_{2,{2;2}}b_{2}}} \\ {= {\frac{1}{3}\begin{pmatrix}  - & - & 1 \end{pmatrix}\begin{pmatrix}  - & 0.01 & 0.02 \end{pmatrix}^{T}}} \\ {= 0.006667} \end{matrix}$ $\begin{matrix} {{h_{3,1}^{s}(0)} = {{\xi_{3}(0)}\Pr\left\{ {BX}_{3,1} \right\}}} \\ {= {{\xi_{3}(0)}\Pr\left\{ {{\frac{r_{3;3}}{r_{3}}A_{3,{1;3}}B_{3}} + {\frac{r_{3;14}}{r_{3}}A_{3,{1;14}}A_{14;D}}} \right\}}} \\ {= {{\frac{1}{3} \times \frac{1}{2}a_{3,{1;3}}b_{3}} + {\frac{1}{2}a_{3,{1;14}}a_{14;D}}}} \\ {= {{\frac{1}{3} \times \frac{1}{2}\begin{pmatrix}  - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.05 \end{pmatrix}} + {\frac{1}{2}\begin{pmatrix}  - & 0.6 \end{pmatrix}\begin{pmatrix}  - \\ 0.1 \end{pmatrix}}}} \\ {= 0.01833} \end{matrix}$

The calculation result of rank probability

${h_{kj}^{r}(y)} = \frac{h_{kj}^{s}(y)}{\sum\limits_{H_{kj} \in {S_{H}{(y)}}}{h_{kj}^{s}(y)}}$

is as follows:

index H_(kj) h_(kj) ^(r) (0) 1 BX_(3, 1) 0.3995 2 BX_(1, 1) 0.3826 3 BX_(2, 2) 0.1453 4 BX_(2, 1) 0.0725

According to 3, based on FIG. 4, one can get S_(12K)(0)=S_(13K)(0)={1}, and S_(4K)(0)S_(7K)(0)S_(9K)(0)={1,2}, and S_(6K)(0)=S_(11K)(0)={2,3}, and S_(5K)(0)=S_(8K)(0)=S_(10K)(0)={1,2,3}. Accordingly, one can get m₁₂(0)=m₁₃(0)=1, and m₄(0)=m₆(0)=m₇(0)=m₉(0)=m₁₁(0)=2, and m₅(0)=m₈(0)=m₁₀(0)=3. Since BX₁ and BX₃ have only one abnormal state indexed by “1”, thus S_(1J)(0)=S_(3J)(0)={1}. And BX₂ has two abnormal states indexed by “1” and “2”, thus S_(2J)(0)={1,2}.

According to 4, assume ω_(kj)=1 (k∈{1,2,3}).

According to 5, with the following formula:

${\rho_{i}(y)} = \left. {\frac{1}{m_{i}^{\prime}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\omega_{kj}{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}}}}}}} \middle| {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right|$

where i∈S_(Xs)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13} and m′_(i)(y)=m_(i)(y). Based on FIG. 4, since ω_(kj)=1, k∈{1,2,3}, thus

$\begin{matrix} {{\rho_{4}(0)} = \left. {\frac{1}{m_{4}(0)}{\sum\limits_{k \in {\{{1,2}\}}}{\sum\limits_{j \in {S_{kJ}{(0)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(0)} \right\}}}}}} \right|} \\ {\left. {{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(0)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(0)} \right\}}} \right|} \\ {= {\frac{1}{2}Pr\left\{ X_{4,1} \right\}\begin{pmatrix} {{\left. {\Pr\left\{ H_{1,1} \right.X_{4,1}} \right\} - {\Pr\left\{ H_{1,1} \right\}}}} \\ {{+ \left. {\Pr\left\{ H_{2,1} \right.X_{4,1}} \right\}} - {\Pr\left\{ H_{2,1} \right\}}} \\ {+ {{{\Pr\left\{ {H_{2,2}❘X_{4,1}} \right\}} - {\Pr\left\{ H_{2,2} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}Pr\left\{ X_{4,1} \right\}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}❘X_{4,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{4,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\begin{pmatrix} {\begin{matrix} {\Pr\left\{ {{BX}_{1,1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\ {{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}} \end{matrix}} \\ {+ {\begin{matrix} {\Pr\left\{ {{BX}_{2,1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\ {{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}} \end{matrix}}} \\ {+ {\begin{matrix} {\Pr\left\{ {{BX}_{2,2}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)} \right\}} \\ {{- \Pr}\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}} \end{matrix}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\begin{pmatrix} {{{\Pr\left\{ {A_{4,{1;1},1}{BX}_{1,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{1,1} \right\}}}} \\ {+ {{{\Pr\left\{ {A_{4,{1;2},1}{BX}_{2,1}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,1} \right\}}}}} \\ {+ {{{\Pr\left\{ {A_{4,{1;2},2}{BX}_{2,2}} \right\}} - {\Pr\left\{ X_{4,1} \right\}\Pr\left\{ {BX}_{2,2} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\begin{pmatrix} {{\left( {{\Pr\left\{ A_{4,{1;1},1} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{1,1} \right\}}} \\ {+ {{\left( {{\Pr\left\{ A_{4,{1;2},1} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,1} \right\}}}} \\ {+ {{\left( {{\Pr\left\{ A_{4,{1;2},2} \right\}} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,2} \right\}}}} \end{pmatrix}}} \end{matrix}$

In which

$\begin{matrix} {{\Pr\left\{ X_{4.1} \right\}} = {\Pr\left\{ {{\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}A_{12;D}} \\ {{+ \frac{r_{1;13}}{r_{1}}}A_{1;13}A_{13;D}} \end{pmatrix}}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}}} \right\}}} \\ {= {\Pr\left\{ {{\frac{1}{2}{A_{4,{1;1}}\ \begin{pmatrix} {\frac{1}{3}A_{1;1}B_{1}} \\ {{+ \frac{1}{3}}A_{1;12}A_{12;D}} \\ {{+ \frac{1}{3}}A_{1;13}A_{13;D}} \end{pmatrix}}} + {\frac{1}{2}A_{4,{1;2}}A_{2;2}B_{2}}} \right\}}} \\ {= {{\frac{1}{2}{a_{4,{1;1}}\ \begin{pmatrix} {\frac{1}{3}a_{1;1}b_{1}} \\ {{+ \frac{1}{3}}a_{1;12}a_{12;D}} \\ {{+ \frac{1}{3}}a_{1;13}a_{13;D}} \end{pmatrix}}} + {\frac{1}{2}a_{4,{1;2}}a_{2;2}b_{2}}}} \\ {= {{\frac{1}{2}\begin{pmatrix}  - & 0.9 \end{pmatrix}\begin{pmatrix} {\frac{1}{3}\begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.08 \end{pmatrix}} \\ {{+ \frac{1}{3}}\begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}\begin{pmatrix}  - \\ 0.1 \end{pmatrix}} \\ {{+ \frac{1}{3}}\begin{pmatrix}  - & - \\  - & 0.8 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \end{pmatrix}} \end{pmatrix}} + {\frac{1}{2}\begin{pmatrix}  - & 0.5 & 0.5 \end{pmatrix}}}} \\ {\begin{pmatrix}  - & - & - \\  - & 1 & - \\  - & - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}} \\ {= 0.0312} \end{matrix}$

Thus:

$\begin{matrix} {{\rho_{4}(0)} = {\frac{1}{2}\begin{pmatrix} {{\left( {a_{4,{1;1},1} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{1,1} \right\}}} \\ {+ {{\left( {a_{4,{1;2},1} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,1} \right\}}}} \\ {+ {{\left( {a_{4,{1;2},2} - {\Pr\left\{ X_{4,1} \right\}}} \right)\Pr\left\{ {BX}_{2,2} \right\}}}} \end{pmatrix}}} \\ {= {{\frac{1}{2}\begin{pmatrix} {{\left( {0.9 - 0.0312} \right)0.05267}} \\ {+ {{\left( {0.5 - 0.0312} \right)0.01}}} \\ {+ {{\left( {0.5 - 0.0312} \right)0.02}}} \end{pmatrix}} = 0.0299}} \end{matrix}$

Similarly,

$\begin{matrix} {{\rho_{5}(0)} = {\frac{1}{m_{5}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{3}\Pr\left\{ X_{5,1} \right\}\begin{pmatrix} {\left. {\Pr\left\{ {BX}_{1,1} \right.X_{5,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\ \left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{5,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{5,1}}} \right\} \\ {{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{5,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}} \end{pmatrix}}} \\ {= 0.030331} \end{matrix}$ $\begin{matrix} {{\rho_{6}(0)} = {\frac{1}{m_{6}(0)}{\sum\limits_{k \in {\{{2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{{1,2}\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{2}\Pr\left\{ X_{6,1} \right\}\begin{pmatrix} {\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{6,1}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{6,1}}} \right\} -} \\ {{{\Pr\left\{ {BX}_{2,2} \right\}}} + {{\Pr\left\{ {{{BX}_{3,1}\left. X_{6,1} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}} \right.}}} \end{pmatrix}}} \\ {{+ \frac{1}{2}}{\Pr\left( X_{6,2} \right\}}\begin{pmatrix} {\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{6,2}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{6,2}}} \right\} -} \\ {{\left. {{{{{\Pr\left\{ {BX}_{2,2} \right\}}} +}}\Pr\left\{ {BX}_{3,1} \right.X_{6,2}} \right\} - {\Pr\left\{ {BX}_{3,1} \right\}}}} \end{pmatrix}} \\ {= 0.040694} \end{matrix}$ $\begin{matrix} {{\rho_{7}(0)} = {\frac{1}{m_{7}(0)}{\sum\limits_{k \in {\{{1,2}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{2}\Pr\left\{ X_{7,1} \right\}\begin{pmatrix} {{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{7,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\ {{{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{7,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}} \\ {{{+ \left. {\Pr\left\{ {BX}_{2,2} \right.X_{7,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}} \end{pmatrix}}} \\ {= 0.020938} \end{matrix}$ $\begin{matrix} {{\rho_{8}(0)} = {\frac{1}{m_{8}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{3}\Pr\left\{ X_{8,1} \right\}\begin{pmatrix} {\left. {\Pr\left\{ {BX}_{1,1} \right.X_{8,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\ \left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{8,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{8,1}}} \right\} \\ {{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{8,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}} \end{pmatrix}}} \\ {= 0.01785} \end{matrix}$ $\begin{matrix} {{\rho_{9}(0)} = {\frac{1}{m_{9}(0)}{\sum\limits_{k \in {\{{1,2}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{2}\Pr\left\{ X_{9,1} \right\}\begin{pmatrix} {{\left. {\Pr\left\{ {BX}_{1,1} \right.X_{9,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\}}}} \\ {{{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{9,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}}}} \\ {{{+ \left. {\Pr\left\{ {BX}_{2,2} \right.X_{9,1}} \right\}} - {\Pr\left\{ {BX}_{2,2} \right\}}}} \end{pmatrix}}} \\ {= 0.020938} \end{matrix}$ $\begin{matrix} {{\rho_{10}(0)} = {\frac{1}{m_{10}(0)}{\sum\limits_{k \in {\{{1,2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{3}\Pr\left\{ X_{10,1} \right\}\begin{pmatrix} {\left. {\Pr\left\{ {BX}_{1,1} \right.X_{10,1}} \right\} - {\Pr\left\{ {BX}_{1,1} \right\} }} \\ \left. {{+ \left. {\Pr\left\{ {BX}_{2,1} \right.X_{10,1}} \right\}} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{10,1}}} \right\} \\ {{- \Pr}\left\{ {BX}_{2,2} \right\}{{{+ \left. {\Pr\left\{ {BX}_{3,1} \right.X_{10,1}} \right\}} - {\Pr\left\{ {BX}_{3,1} \right\}}}}} \end{pmatrix}}} \\ {= 0.021232} \end{matrix}$ $\begin{matrix} {{\rho_{11}(0)} = {\frac{1}{m_{11}(0)}{\sum\limits_{k \in {\{{2,3}\}}}^{\;}\;{\sum\limits_{j \in {S_{kJ}{(0)}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{ig}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {= {\frac{1}{2}\Pr\left\{ X_{11,1} \right\}\begin{pmatrix} {\left. {\left. {\Pr\left\{ {BX}_{2,1} \right.X_{11,1}} \right\} - {\Pr\left\{ {BX}_{2,1} \right\}{ + }\Pr\left\{ {BX}_{2,2} \right.X_{11,1}}} \right\} -} \\ {{\left. {\Pr\left\{ {BX}_{2,2} \right\}{ + }\Pr\left\{ {BX}_{3,1} \right.X_{11,1}} \right\} - {\Pr\left\{ {BX}_{3,1} \right\}}}} \end{pmatrix}}} \\ {= 0.027049} \end{matrix}$ $\begin{matrix} {{\rho_{12}(0)} = {\frac{1}{m_{12}(0)}{\sum\limits_{k \in {\{ 1\}}}^{\;}\;{\sum\limits_{j \in {\{ 1\}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{12g}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {{= {{\frac{1}{1}\Pr\left\{ X_{12,1} \right\}\left. {\Pr\left\{ {BX}_{1,1} \right.X_{12,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}}} \\ {= 0.063} \end{matrix}$ $\begin{matrix} {{\rho_{13}(0)} = {\frac{1}{m_{13}(0)}{\sum\limits_{k \in {\{ 1\}}}^{\;}\;{\sum\limits_{j \in {\{ 1\}}}^{\;}\;{\sum\limits_{g \in {\{ 1\}}}^{\;}\;{\Pr\left\{ {X_{13g}\left. {E(0)} \right\}} \right.\Pr\left\{ {{H_{kj}\left. {X_{ig}{E(0)}} \right\}} -} \right.}}}}}} \\ {\Pr\left\{ {H_{kj}\left. {E(0)} \right\}} \right.} \\ {{= {{\frac{1}{1}\Pr\left\{ X_{13,1} \right\}\left. {\Pr\left\{ {BX}_{1,1} \right.X_{13,1}} \right\}} - {\Pr\left\{ {BX}_{1,1} \right\}}}}} \\ {= 0.00792} \end{matrix}$

According to 7, take

${{\rho_{i}(y)} = {\max\limits_{l \in {S_{Xs}{(y)}}}\left\{ {\rho_{l}(y)} \right\}}},$

one can get the following:

ρ₁₅(0)=ρ₁₆(0)=max{0.0299, 0.030331, 0.040694, 0.020938, 0.01785, 0.020838, 0.021232, 0.027049, 0.063, 0.00792}=0.063.

And then:

i ρ_(i)(0) 4 0.0299 5 0.030331 6 0.040694 7 0.020938 8 0.01785 9 0.020938 10 0.021232 11 0.027049 12 0.063 13 0.00792 15 0.063 16 0.063 According to 6, let β_(i)=1/α_(i) and assign their values as follows:

i α_(i) β_(i) 4 100 0.01 5 2 0.5 6 50 0.02 7 1 1 8 2 0.5 9 2 0.5 10 2 0.5 11 1 1 12 1 1 13 1 1 15 1 1 16 1 1 According to 3, assume n=1, given m_(i)(0), λ_(i)(0) is calculated as follows:

i m_(i)(0) λ_(i)(0) 4 2 1/2 5 3 1/3 6 2 1/2 7 1 1 8 3 1/3 9 1 1 10 1 1 11 1 1 12 1 1 13 1 1 15 1 1 16 1 1 In which, according to 7, let λ₁₅(0)=λ₁₆(0)=1. According to 8, take

${{I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\Sigma_{i \in S_{X^{(y)}}}{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}},$

we can obtain the following results:

${I_{4}(0)} = {\frac{{\lambda_{4}(0)}\beta_{4}{\rho_{4}(0)}}{\sum\limits_{i \in {\{{4,\ldots,13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{2} \times 0.01 \times 0.0299}{{0.2}7457861} = {{0.0}00544}}}$ ${I_{5}(0)} = {\frac{{\lambda_{5}(0)}\beta_{5}{\rho_{5}(0)}}{\sum\limits_{i \in {\{{4,\ldots,13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{3} \times {0.5} \times {0.0}30331}{{0.2}7457861} = {{0.0}18411}}}$ ${I_{6}(0)} = {\frac{{\lambda_{6}(0)}\beta_{6}{\rho_{6}(0)}}{\sum\limits_{i \in {\{{{4\text{,…,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{2} \times {0.0}2 \times 0.040694}{{0.2}7457861} = {{0.0}01482}}}$ ${I_{7}(0)} = {\frac{{\lambda_{7}(0)}\beta_{7}{\rho_{7}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,1}3},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.020938}{{0.2}7457861} = {{0.0}76255}}}$ ${I_{8}(0)} = {\frac{{\lambda_{8}(0)}\beta_{8}{\rho_{8}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{\frac{1}{3} \times 0.5 \times 0.01785}{{0.2}7457861} = {{0.0}10835}}}$ ${I_{9}(0)} = {\frac{{\lambda_{9}(0)}\beta_{9}{\rho_{9}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 0.5 \times 0.020938}{027457861} = {{0.0}38128}}}$ ${I_{10}(0)} = {\frac{{\lambda_{10}(0)}\beta_{10}{\rho_{10}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 0.5 \times 0.021232}{0.27457861} = {{0.0}38663}}}$ ${I_{11}(0)} = {\frac{{\lambda_{11}(0)}\beta_{11}{\rho_{11}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,1}3},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.027049}{{0.2}7457861} = {{0.0}98511}}}$ ${I_{12}(0)} = {\frac{{\lambda_{12}(0)}\beta_{12}{\rho_{12}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$ ${I_{13}(0)} = {\frac{{\lambda_{13}(0)}\beta_{13}{\rho_{13}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.00792}{{0.2}7457861} = {{0.0}28844}}}$ ${I_{15}(0)} = {\frac{{\lambda_{15}(0)}\beta_{15}{\rho_{15}(0)}}{\sum\limits_{i \in {\{{4,\text{...},13,15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$ ${I_{16}(0)} = {\frac{{\lambda_{16}(0)}\beta_{16}{\rho_{16}(0)}}{\sum\limits_{i \in {\{{4,{\text{...,}13},15,16}\}}}{{\lambda_{i}(0)}\beta_{i}{\rho_{i}(0)}}} = {\frac{1 \times 1 \times 0.063}{{0.2}7457861} = {{0.2}29442}}}$

According to 9, the ranking results of X_(i) are:

index i I_(i)(0) 1 12 0.229442 2 15 0.229442 3 16 0.229442 4 11 0.098511 5 7 0.076255 6 10 0.038663 7 9 0.038128 8 13 0.028844 9 5 0.018411 10 8 0.010835 11 6 0.001482 12 4 0.000544

Take the first 5 X-type variables into detection. Assuming the detection results are X_(7,1), X_(11,0), X_(12,1), X_(15,0) and X_(16,0), FIG. 4 changes as FIG. 8, in which, X₁₅ and X₁₆ are cause-specific variables for BX_(3,1) and all the detection results are negative (variables are all at state 0), thus we know that the state of BX₃ is BX_(3,0).

According to the parameters in Example 1, FIG. 8 is simplified as FIG. 9 by using the aforementioned simplification rules 2, 3 and 5, and E(1)=E⁺(0)E(0)=E⁺(0)=X_(7,1)X_(11,0)X_(12,1). Based on E(1) and similar to example 1, FIG. 9 can be divided and simplified as FIG. 10 and FIG. 11 according to the DUCG algorithms in [8].

According to the algorithms in [4]-[9], calculate the probabilities ζ_(i)(y)=ζ_(i)(1) of the sub-graph FIG. 9 and FIG. 10:

Based on FIG. 10,

$\begin{matrix} {{\zeta_{1}(1)} = {\Pr\left\{ {E(1)} \right\}}} \\ {= {\Pr\left\{ {X_{7,1}X_{{11},0}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {X_{7,1}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}{A_{4;1}\left( {{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{{1;12},1}X_{12,1}}} \right)}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {A_{7,{1;4}}{A_{4;1}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {A_{7,{1;4}}{A_{4;1}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)}A_{{12},{1;D}}} \right\}}} \\ {= {a_{7,{1;4}}{a_{4;1}\left( {{\frac{1}{2}a_{1;1}b_{1}} + {\frac{1}{2}a_{{1;12},1}}} \right)}a_{12,{1;D}}}} \\ {= {\left( {- 0.7} \right)\begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}\left( {{\frac{1}{2}\ \begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.08 \end{pmatrix}} + {\frac{1}{2}\begin{pmatrix}  - \\ 0.7 \end{pmatrix}}} \right)0.1}} \\ {= {0.02457}} \end{matrix}$

Based on FIG. 11,

$\begin{matrix} {{\zeta_{2}(1)} = {\Pr\left\{ {E(1)} \right\}}} \\ {= {\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}} \right)\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}} \right)A_{12,{1;D}}} \right\}}} \\ {= {\Pr\left\{ {\left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;2}} \right)*\left( {\frac{1}{1}A_{11,{0;6}}\frac{1}{1}A_{6;2}} \right)A_{2;2}B_{2}A_{12,{1;D}}} \right\}}} \\ {= {\left( {a_{7,{1;4}}a_{4;2}} \right)*\left( {a_{11,{0;6}}a_{6;2}} \right)a_{2;2}b_{2}a_{12,{1;D}}}} \\ {= {\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.5 & 0.5 \end{pmatrix}} \right)*\left( \begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix} \right.}} \\ {\left. \begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix} \right)\begin{pmatrix}  - & - & - \\  - & 1 & - \\  - & - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1} \\ {= 0.00042} \end{matrix}$

In which, the original parameter a_(11,0;6)=(- - -) is modified as a_(11,0;6)=(1 1-0.3 1-0.8), because X_(11,0) is a negative evidence, which means none of the abnormal states occurs and X_(11,0)=1−X_(11,1) according to [4].

According to the algorithms in [8], the weighting coefficients

${\xi_{i}(y)} = \frac{\zeta_{i}(y)}{\sum\limits_{j}{\zeta_{j}(y)}}$

of the sub-graphs are:

${\xi_{1}(1)} = {\frac{\zeta_{1}(1)}{{\zeta_{1}(1)} + {\zeta_{2}(1)}} = {\frac{{0.0}2457}{{{0.0}2457} + {{0.0}0042}} = {{0.9}832}}}$ ${\xi_{2}(1)} = {\frac{\zeta_{2}(1)}{{\zeta_{1}(1)} + {\zeta_{2}(1)}} = {\frac{{0.0}0042}{{{0.0}2457} + {0.00042}} = {{0.0}1681}}}$

According to 1 and based on the DUCG algorithms in [4]-[9], when y=0+1=1, the state probability h_(kj) ^(s)(y)=h_(kj) ^(s)(1) of H_(kj) is:

$\begin{matrix} {{h_{1,1}^{s}(1)} = {{\xi_{1}(1)}\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}} \\ {= {{\xi_{1}(1)}\Pr\left\{ {{BX}_{1,1}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\ {= {{\xi_{1}(1)}\frac{\Pr\left\{ {{BX}_{1,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}{\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}}}} \\ {= {0.983193 \times \frac{0.00022932}{0.0003574}}} \\ {= {0.983193 \times 0.64163}} \\ {= 0.63085} \end{matrix}$

In which

${\Pr\left\{ {X_{7,1}X_{11,0}X_{12,1}} \right\}} = {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4;1}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}X_{12,1}} \right\}} = {{\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{{4;1},1}{BX}_{1,1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \\ {{+ {{BX}_{1,1}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right){BX}_{2}A_{12,{1;D}}} \end{Bmatrix}} = {{\Pr\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{A_{{4;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\ {{+ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}} \end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)} \end{pmatrix}\frac{r_{2;2}}{r_{2}}A_{2;2}B_{2}A_{12,{1;D}}} \right\}} = {{\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{a_{{4;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\ {{+ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;2}}{r_{4}}a_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}} = {{\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{a_{{4;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\ {{+ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;2}}{r_{4}}a_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}\; = {{\begin{pmatrix} {\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\frac{1}{2}\begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}\left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \\ {{+ \left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)}\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\frac{1}{2}\begin{pmatrix}  - & - & - \\  - & 0.5 & 0.5 \end{pmatrix}} \right)*\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = {{\left( {{0.12285\begin{pmatrix}  - & 0.5 & 0.35 \end{pmatrix}} + {0.39\begin{pmatrix}  - & 0.175 & 0.175 \end{pmatrix}*\begin{pmatrix}  - & 0.5 & 0.35 \end{pmatrix}}} \right)\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = {{\begin{pmatrix}  - & 0.09555 & 0.066885 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = \; 0.0003574}}}}}}}}$

${\Pr\left\{ {{BX}_{1,1}X_{7,1}X_{11,0}X_{12,1}} \right\}} = {{\Pr\left\{ {{BX}_{1,1}\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4;1}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4;2}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}\; = {{\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{{4;1},1}{BX}_{1,1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \\ {{+ {{BX}_{1,1}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)}}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right){BX}_{2}A_{12,{1;D}}} \end{Bmatrix}}\; = {{\Pr\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{A_{{4;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\ {{+ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1,{1;1},1}B_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}} \end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)} \end{pmatrix}\frac{r_{2;2}}{r_{2}}A_{2;2}B_{2}A_{12,{1;D}}} \right\}}\; = {{\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;1}}{r_{4}}{a_{{4;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\ {{+ \begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1},1}b_{1,1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12},1}} \end{pmatrix}}\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4}}\frac{r_{4;2}}{r_{4}}a_{4;2}} \right)*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}\; = {{\begin{pmatrix} {\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\frac{1}{2}\begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}\left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \\ {{+ \left( {{\frac{1}{2} \times 0.08} + {\frac{1}{2} \times 0.7}} \right)}\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\frac{1}{2}\begin{pmatrix}  - & - & - \\  - & 0.5 & 0.5 \end{pmatrix}} \right)*\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = {{\left( {{0.12285\begin{pmatrix}  - & 0.5 & 0.35 \end{pmatrix}} + {0.39\begin{pmatrix}  - & 0.175 & 0.175 \end{pmatrix}*\begin{pmatrix}  - & 0.5 & 0.35 \end{pmatrix}}} \right)\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = \;{{\begin{pmatrix}  - & 0.09555 & 0.066885 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}\; = \; 0.00022932}}}}}}}$ ${{{In}\mspace{14mu}{the}\mspace{14mu}{same}\mspace{14mu}{way}},\begin{matrix} {{h_{2,1}^{s}(1)} = {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}} \\ {= {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,1}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\ {= 0.007} \end{matrix}}\mspace{14mu}$

$\begin{matrix} {{h_{2,2}^{s}(1)} = {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}} \\ {= {{\xi_{2}(1)}\Pr\left\{ {{BX}_{2,2}❘{X_{7,1}X_{11,0}X_{12,1}}} \right\}}} \\ {= 0.009781} \end{matrix}$

Based on

${{h_{kj}^{r}(1)} = \frac{h_{kj}^{s}(1)}{\sum\limits_{H_{kj} \in {S_{H}{(1)}}}{h_{kj}^{s}(1)}}},$

the rank probabilities are:

index H_(kj) h_(kj) ^(r) (1) 1 BX_(1, 1) 0.974089 2 BX_(2, 2) 0.015103 3 BX_(2, 1) 0.010809 Compared to the rank probabilities h_(kj) ^(r)(0) before detections, we see (1) BX_(3,1) which ranks first is eliminated and the possible result is a reduced state space S_(H)(1), and (2) the rank probability of BX_(1,1) is far bigger than the rest two, which means that the cause of abnormality in the real target system can be verified as BX_(1,1) with only five X-type variables being detected.

Example 2: y=1

According to 2, when no evidence (y=0), based on FIG. 4, S_(H)(0)={H_(1,1), H_(2,1), H_(2,2), H_(3,1)}={BX_(1,1), BX_(2,1), BX_(2,2), BX_(3,1)}, and the state pending X-type variables are {X₄, X₅, X₆, X₇, X₈, X₉, X₁₀, X₁₁, X₁₂, X₁₃, X₁₅, X₁₆}. All these states are unknown and single-connect to H_(k) in S_(H)(0) without state-known blocking variables, so that S_(X)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16}. Since X₁₅ and X₁₆ are cause-specific variable for BX_(3,1), they are eliminated from S_(X)(0), that means S_(Xs)(0)={4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. Accordingly, S_(4G)(0)=S_(5G)(0)=S_(7G)(0)=S_(8G)(0)=S_(9G)(0)=S_(10G)(0)=S_(11G)(0)=S_(12G)(0)=S_(13G)(0)={1} and S_(6G)(0)={1, 2}.

As example 1 and based on FIG. 9, E(1)=X_(7,1)X_(11,0)X_(12,1), S_(H)(1)={H_(1,1), H_(2,1), H_(2,2)}={BX_(1,1), BX_(2,1), BX_(2,2)} and the testing X variables are {X₄, X₅, X₆, X₈, X₉, X₁₀}. All of them are states unknown and single-connect to like H_(k)∈{BX₁, BX₂} without state-known blocking variables. According to 9, X₄ is the only upstream variable of X_(7,1) and can be eliminated from the ranking, thus S_(X)(1)={5, 6, 8, 9, 10}. Since there is no cause-specific variable, then S_(Xs)(1)=S_(X)(1). As the same as y=0, one can get S_(4G)(1)=S_(5G)(1)=S_(8G)(1)=S_(9G)(1)=S_(10G)(1)={1} and S_(6G)(1)={1, 2}.

According to 3 and based on FIG. 9, one can get the following equations: S_(13K)(1)={1}, S_(4K)(1)=S_(5K)(1)=S_(8K)(1)=S_(9K)(1)=S_(10K)(1)={1,2}, and S_(6K)(1)={2}; and accordingly m₆(1)=1, m₄(1)=m₅(1)=m₈(1)=m₁₀(1) m₉(1)=2, S_(1J)(1)={1}, and S_(2J)(1)={2}. Still same as y=0, ω_(kj)=1 (k∈{1,2}).

According to 5, take the following formula for calculation as same as in example 1:

${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\sum\limits_{j \in {S_{kJ}{(y)}}}{\omega_{kj}{\sum\limits_{g \in {S_{iG}{(y)}}}{\Pr\left\{ X_{ig} \middle| {E(y)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {\Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}}$

where i∈S_(Xs)(0)={4, 5, 6, 8, 9, 10}. Note m′_(i)(y)=m_(i)(y) in this example.

According to FIG. 9, since ω_(kj)=1 and k∈{1, 2}, we have

$\begin{matrix} {{\rho_{4}(1)} = {\frac{1}{m_{4}(1)}{\sum\limits_{k \in {\{{1,2}\}}}{\sum\limits_{j \in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{\Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {\Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{2}\Pr\left\{ X_{4,1} \middle| {E(1)} \right\}\begin{pmatrix} {{{\Pr\left\{ H_{1,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{1,1} \middle| {E(1)} \right\}}}} \\ {+ {{{\Pr\left\{ H_{2,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{2,1} \middle| {E(1)} \right\}}}}} \\ {+ {{{\Pr\left\{ H_{2,2} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ H_{2,2} \middle| {E(1)} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\Pr\left\{ X_{4,1} \middle| {E(1)} \right\}\begin{pmatrix} {{{\Pr\left\{ {BX}_{1,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{1,1} \middle| {E(1)} \right\}}}} \\ {+ {{{\Pr\left\{ {BX}_{2,1} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{2,1} \middle| {E(1)} \right\}}}}} \\ {+ {{{\Pr\left\{ {BX}_{2,2} \middle| {X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {BX}_{2,2} \middle| {E(1)} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\frac{1}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}{XX}_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{1,1} \middle| {E(1)} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}{XX}_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{2,1} \middle| {E(1)} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}{XX}_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ B_{2,2} \middle| {E(1)} \right\}}}}} \end{pmatrix}}} \end{matrix}$

In which

$\begin{matrix} {{\Pr\left\{ {X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {X_{4,1}X_{7,1}X_{{11},0}X_{{12},1}} \right\}}} \\ {= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{{11},0}X_{{12},1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{{11},0}X_{12,1}} \right\}}}} \\ {= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{ó}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}BX_{1}\frac{r_{11;ó}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}BX_{2}A_{12,{1;D}}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)BX_{2}A_{12,{1;D}}} \end{Bmatrix}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)} \end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2;2} B_{2} A_{12,{1;D}}} \right\}}} \\ {= {\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1;12}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\ {= {\begin{pmatrix} {\left( {0.7\frac{1}{2}\begin{pmatrix}  - & 0.9 \end{pmatrix}\begin{pmatrix}  - \\ 0.39 \end{pmatrix}} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \\ {{+ \left( {0.7\frac{1}{2}\begin{pmatrix}  - & 0.5 & 0.5 \end{pmatrix}} \right)}*\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}} \\ {= 0.00035742} \end{matrix}$

$\begin{matrix} {{\Pr\left\{ {{BX}_{1,1}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{1,1}X_{4,1}X_{7,1}X_{{11},0}X_{{12},1}} \right\}}} \\ {= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{{11},0}X_{{12},1}{BX}_{1,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{{11},0}X_{12,1}{BX}_{1,1}} \right\}}}} \\ {= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1},1}{BX}_{1,1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}BX_{2}{BX}_{1,1}}} \right)}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1},1}BX_{1,1}\frac{r_{11;6}}{r_{11}}A_{{11},{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}BX_{2}A_{12,{1;D}}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{ó}}A_{6;2}} \right)BX_{2}{BX}_{1,1}A_{12,{1;D}}} \end{Bmatrix}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1,{1;12}}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}} \right)\begin{pmatrix} {{\frac{r_{1;1}}{r_{1}}A_{1,{1;1}}B_{1}} +} \\ {\frac{r_{1;12}}{r_{1}}A_{1,{1;12}}} \end{pmatrix}} \end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2;2} B_{2} A_{12,{1;D}}} \right\}}} \\ {= {\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{4}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1},1}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1,{1;1}}b_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1,{1;12}}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{6;2}} \right)\begin{pmatrix} {{\frac{r_{1;1}}{r_{1}}a_{1,{1;1}}b_{1}} +} \\ {\frac{r_{1;12}}{r_{1}}a_{1,{1;12}}} \end{pmatrix}} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\ {= {\begin{pmatrix} {\left( {0.7\frac{1}{2}0.9(0.39)} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)} \\ {{+ \left( {0.7\frac{1}{2}\begin{pmatrix}  - & 0.5 & 0.5 \end{pmatrix}} \right)}*\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.6 & 0.3 \\  - & 0.4 & 0.7 \end{pmatrix}} \right)(0.39)} \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}} \\ {= 0.0002293} \end{matrix}$

$\begin{matrix} {{\Pr\left\{ {{BX}_{2,1}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{2,1}X_{4,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\ {= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}}}} \\ {= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4}}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}{BX}_{2,1}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}{BX}_{2,1}A_{12,{1;D}}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \right){BX}_{2,1}A_{112,{1;D}}} \end{Bmatrix}}} \\ {= {P r\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{1;12}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}} \right)} \end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2,{1;2}} B_{2} A_{12,{1;D}}} \right\}}} \\ {= {\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{1;12}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},1}} \right)} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},1}} \right)}\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},1}} \right)} \end{pmatrix}a_{2,{1;2}}b_{2}a_{12,{1;D}}}} \\ {= {\begin{pmatrix} {\left( {0.7\frac{1}{2}\begin{pmatrix}  - & 0.9 \end{pmatrix}\begin{pmatrix}  - \\ 0.39 \end{pmatrix}} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - \\ 0.6 \\ 0.4 \end{pmatrix}} \right)} \\ {{+ \left( {0.7\frac{1}{2}(0.5)} \right)}\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - \\ 0.6 \\ 0.4 \end{pmatrix}} \right)} \end{pmatrix}(0.01)0.1}} \\ {= 0.0001489} \end{matrix}$

$\begin{matrix} {{\Pr\left\{ {{BX}_{2,2}X_{4,1}{E(1)}} \right\}} = {\Pr\left\{ {{BX}_{2,2}X_{4,1}X_{7,1}X_{11,0}X_{12,1}} \right\}}} \\ {= {{\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,1}} \right\}} = {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}X_{4,1}X_{11,0}X_{12,1}{BX}_{2,2}} \right\}}}} \\ {= {\Pr\left\{ {\frac{r_{7;4}}{r_{7}}{A_{7,{1;4},1}\left( {{\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}} + {\frac{r_{4;2}}{r_{4}}A_{4,{1;2}}{BX}_{2}}} \right)}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{6;2}{BX}_{2}A_{12,{1;D}}{BX}_{2,2}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}A_{4,{1;1}}{BX}_{1}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},1}{BX}_{2,2}A_{12,{1;D}}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \right){BX}_{2,2}A_{112,{1;D}}} \end{Bmatrix}}} \\ {= {P r\left\{ {\begin{pmatrix} {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{A_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}A_{{1;12},1}} \end{pmatrix}}\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}A_{11,{0;6}}\frac{r_{6;2}}{r_{6}}A_{{6;2},2}} \right)} \end{pmatrix}\frac{r_{2;2}}{r_{2}} A_{2,{2;2}} B_{2} A_{12,{1;D}}} \right\}}} \\ {= {\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;1}}{r_{4}}{a_{4,{1;1}}\begin{pmatrix} {\frac{r_{1;1}}{r_{1}}a_{1;1}b_{1}} \\ {{+ \frac{r_{1;12}}{r_{1}}}a_{{1;12},1}} \end{pmatrix}}} \right)\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},2}} \right)} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}a_{7,{1;4},1}\frac{r_{4;2}}{r_{4}}A_{4,{1;2},2}} \right)}*\left( {\frac{r_{11;6}}{r_{11}}a_{11,{0;6}}\frac{r_{6;2}}{r_{6}}a_{{6;2},2}} \right)} \end{pmatrix}a_{2,{2;2}}b_{2}a_{12,{1;D}}}} \\ {= {\begin{pmatrix} {\left( {0.7\frac{1}{2}\begin{pmatrix}  - & 0.9 \end{pmatrix}\begin{pmatrix}  - \\ 0.39 \end{pmatrix}} \right)\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - \\ 0.3 \\ 0.7 \end{pmatrix}} \right)} \\ {{+ \left( {0.7\frac{1}{2}(0.5)} \right)}*\left( {\begin{pmatrix} 1 & {1 - 0.3} & {1 - 0.8} \end{pmatrix}\begin{pmatrix}  - \\ 0.3 \\ 0.7 \end{pmatrix}} \right)} \end{pmatrix}(0.02)0.1}} \\ {= 0.000208} \end{matrix}$

Thus

$\begin{matrix} {{\rho_{4}(1)} = {\frac{1}{2}\frac{1}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{4,1}{E(1)}} \right\}} - {\Pr\left\{ {X_{4,1}{E(1)}} \right\}\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \end{pmatrix}}} \\ {= {\frac{1}{2}\frac{1}{0.0003574}\begin{pmatrix} {{0.0002293 - {0.00035742 \times 0.64163}}} \\ {+ {{0.0001489 - {0.00035742 \times 0.41662}}}} \\ {+ {{0.000208 - {0.00035742 \times 0.58286}}}} \end{pmatrix}}} \\ {= 0} \end{matrix}$

Similarly,

$\begin{matrix} {{\rho_{5}(1)} = {\frac{1}{m_{5}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{2}Pr\left\{ X_{5,1} \middle| {E(1)} \right\}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{5,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \end{pmatrix}}} \\ {= 0.092519} \end{matrix}$ $\begin{matrix} {{\rho_{6}(1)} = {\frac{1}{m_{6}(1)}{\sum\limits_{k \in {{\{ 2\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{{1,2}\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{1}Pr\left\{ X_{6,1} \middle| {E(1)} \right\}\left( {{{{\Pr\left\{ {{BX}_{2,1}X_{6,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}} + {{{\Pr\left\{ {{BX}_{2,2}X_{6,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \right)}} \\ {{+ \frac{1}{1}}Pr\left\{ X_{6,2} \middle| {E(1)} \right\}\left( {{{{\Pr\left\{ {{BX}_{2,1}X_{6,2}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}} + {{{\Pr\left\{ {{BX}_{2,2}X_{6,2}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \right)} \\ {= 0.233261} \end{matrix}$ $\begin{matrix} {{\rho_{8}(1)} = {\frac{1}{m_{8}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{2}Pr\left\{ X_{8,1} \middle| {E(1)} \right\}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{8,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \end{pmatrix}}} \\ {= 0.065047} \end{matrix}$ $\begin{matrix} {{\rho_{9}(1)} = {\frac{1}{m_{9}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{2}Pr\left\{ X_{9,1} \middle| {E(1)} \right\}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{9,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \end{pmatrix}}} \\ {= 0.000199} \end{matrix}$ $\begin{matrix} {{\rho_{10}(1)} = {\frac{1}{m_{10}(1)}{\sum\limits_{k \in {{\{{1,2}\}}j}}{\sum\limits_{\in {S_{kJ}{(1)}}}{\sum\limits_{g \in {\{ 1\}}}{Pr\left\{ X_{ig} \middle| {E(1)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(1)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(1)} \right\}}}}}}}}}} \\ {= {\frac{1}{2}\frac{\Pr\left\{ X_{10,1} \middle| {E(1)} \right\}}{\Pr\left\{ {E(1)} \right\}}\begin{pmatrix} {{{\Pr\left\{ {{BX}_{1,1}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{1,1}❘{E(1)}} \right\}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,1}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,1}❘{E(1)}} \right\}}}}} \\ {+ {{{\Pr\left\{ {{BX}_{2,2}X_{10,1}{E(1)}} \right\}} - {\Pr\left\{ {{BX}_{2,2}❘{E(1)}} \right\}}}}} \end{pmatrix}}} \\ {= 0.06229} \end{matrix}$

The calculation results are listed in the following:

i ρ_(i)(1) 4 0 5 0.092519 6 0.233261 8 0.065047 9 0.000199 10 0.06299 According to 6, the same as example 1. assume β_(i)=1/α_(i) and the values are follows:

i α_(i) β_(i) 4 100 0.01 5 2 0.5 6 50 0.02 8 2 0.5 9 2 0.5 10 2 0.5

Given m_(i)(1), λ_(i)(1) is calculated as follows:

i m_(i)(1) λ_(i)(1) 4 2 1/2 5 2 1/2 6 1 1/1 8 3 1/3 9 1 1 10 1 1

According to 8, take

${{I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in S_{X{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}},$

we can obtain the following results:

${I_{4}(1)} = {\frac{{\lambda_{4}(1)}\beta_{4}{\rho_{4}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{2} \times {0.0}1 \times 0}{{0.0}70231} = 0}}$ ${I_{5}(1)} = {\frac{{\lambda_{5}(1)}\beta_{5}{\rho_{5}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{2} \times 0.5 \times 0.0231298}{0.070231} = {{0.3}29338}}}$ ${I_{6}(1)} = {\frac{{\lambda_{6}(1)}\beta_{6}{\rho_{6}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times {0.0}2 \times 0.0046652}{{0.0}70231} = {{0.0}66427}}}$ ${I_{8}(1)} = {\frac{{\lambda_{8}(1)}\beta_{8}{\rho_{8}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{\frac{1}{3} \times 0.5 \times {0.0}10841}{{0.0}70231} = {{0.1}54363}}}$ ${I_{9}(1)} = {\frac{{\lambda_{9}(1)}\beta_{9}{\rho_{9}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times {0.5} \times 0.0000995}{{0.0}70231} = {{0.0}01417}}}$ ${I_{10}(1)} = {\frac{{\lambda_{10}(1)}\beta_{10}{\rho_{10}(1)}}{\sum\limits_{i \in {\{{4,\cdots\mspace{14mu},9,10}\}}}{{\lambda_{i}(1)}\beta_{i}{\rho_{i}(1)}}} = {\frac{1 \times 0.5 \times 0.031495}{{0.0}70231} = {{0.4}48449}}}$

The ranking results are:

index i I_(i)(1) 1 10 0.448449 2 5 0.329338 3 8 0.154363 4 6 0.066427 5 9 0.001417 6 4 0

According to 9, since X₅ is the only upstream variable of X₁₀, X₅ can be eliminated from the ranking. Since the rank probability of X₄ is equal to zero and there is no meaning to detect X₄, thus X₄ can be eliminated from the ranking. So the above ranking becomes:

index i I_(i)(1) 1 10 0.448449 2 8 0.154363 3 6 0.066427 4 9 0.001417

Take the first three X-type variables into detection. Assume the results are X_(8,1), and X_(6,0) respectively. Then FIG. 9 becomes FIG. 12. Since a_(8,1;6,0)=a_(11,0;6,0)=“-”, which means no corresponding causal relationship, FIG. 12 is simplified as FIG. 13 based on the simplification rules, where E(2)=E⁺(1)E(1)=X_(10,1)X_(8,1)X_(6,0)X_(7,1)X_(8,1)X_(10,1)X_(12,1). To calculate h_(1,1) ^(r)(2), h_(2,1) ^(r)(2), and h_(2,2) ^(r)(2), FIG. 13 can be divided and simplified as FIG. 14 and FIG. 15. According to the DUCG algorithms in [8], we can obtain that:

Based on FIG. 14,

$\begin{matrix} {{\zeta_{1}(2)} = {{\Pr\left\{ {E(2)} \right\}} = {Pr\left\{ {X_{7,1}X_{8,1}X_{10,1}X_{12,1}} \right\}}}} \\ {= {\Pr\left\{ {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}} \right)\begin{pmatrix} {\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}BX_{1}} \\ {{+ \frac{r_{8;5}}{r_{8}}}A_{8,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}BX_{1}} \end{pmatrix}\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}BX_{1}} \right)X_{12,1}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)*\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right){BX}_{1}X_{12,1}} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right){BX}_{1}X_{12,1}} \end{Bmatrix}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)*\left( {\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}\frac{r_{5;1}}{r_{5}}A_{5;1}} \right)} \\ {{+ \left( {\frac{r_{7;4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;1}}{r_{4}}A_{4;1}} \right)}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*\frac{r_{10;5}}{r_{10}}A_{10,{1;5}}} \right)} \end{pmatrix}{BX}_{1}X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{2}A_{8,{1;4}}\frac{1}{1}A_{4;1}} \right)*\left( {\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;1}} \right)} \\ {{+ \left( {\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}*\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;1}} \right)} \end{pmatrix}\left( {{\frac{r_{1;1}}{r_{1}}A_{1;1}B_{1}} + {\frac{r_{1;12}}{r_{1}}A_{{1;12},1}X_{12,1}}} \right)X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;1}} \right)*\left( {A_{10,{1;5}}A_{5;1}} \right)} \\ {{+ \left( {A_{7,{1;4}}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}*A_{10,{1;5}}A_{5;1}} \right)} \end{pmatrix}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)X_{12,1}} \right\}}} \\ {= {\Pr\left\{ {\begin{pmatrix} {\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;1}} \right)*\left( {A_{10,{1;5}}A_{5;1}} \right)} \\ {{+ \left( {A_{7,{1;4}}A_{4;1}} \right)}*\left( {\frac{1}{2}A_{8,{1;5}}A_{5;1}*A_{10,{1;5}}} \right)} \end{pmatrix}\left( {{\frac{1}{2}A_{1;1}B_{1}} + {\frac{1}{2}A_{{1;12},1}}} \right)A_{12,{1;D}}} \right\}}} \\ {= {\begin{pmatrix} {\left( {a_{7,{1;4}}*\frac{1}{2}a_{8,{1;4}}a_{4;1}} \right)*\left( {a_{10,{1;5}}a_{5;1}} \right)} \\ {{+ \left( {a_{7,{1;4}}a_{4;1}} \right)}*\left( {\frac{1}{2}a_{8,{1;5}}*a_{10,{1;5}}a_{5;1}} \right)} \end{pmatrix}\left( {{\frac{1}{2}a_{1;1}b_{1}} + {\frac{1}{2}a_{{1;12},1}}} \right)a_{12,{1;D}}}} \\ {= {\begin{pmatrix} {\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}*\frac{1}{2}\begin{pmatrix}  - & 0.8 \end{pmatrix}\begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}} \right)*\left( \left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}} \right) \right)} \\ {{+ \left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - \\  - & 0.9 \end{pmatrix}} \right)}*\left( {\frac{1}{2}\begin{pmatrix}  - & 0.7 \end{pmatrix}*\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - \\  - & 0.7 \end{pmatrix}} \right)} \end{pmatrix}\left( {{\frac{1}{2}\begin{pmatrix}  - & - \\  - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.08 \end{pmatrix}} + {\frac{1}{2}\begin{pmatrix}  - \\ 0.7 \end{pmatrix}}} \right)0.1}} \\ {= {\left( {\left( {{- \ {0.1}}2348} \right) + \left( {{- \ {0.1}}08045} \right)} \right)\begin{pmatrix}  - \\ 0.39 \end{pmatrix}0.1}} \\ {= {0.009029}} \end{matrix}$

Based on FIG. 15,

$\begin{matrix} {{\zeta_{2}(2)} = {{\Pr\left\{ {E(2)} \right\}} = {Pr\left\{ {X_{6,0}X_{7,1}X_{8,1}X_{10,1}X_{12,1}} \right\}}}} \\ {= {\Pr\left\{ {\frac{r_{6;2}}{r_{6}}A_{6,{0;2}}BX_{2}\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}B{X_{2}\begin{pmatrix} {\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}BX_{2}} \\ {{+ \frac{r_{8;5}}{r_{8}}}A_{8,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}{BX}_{2}} \end{pmatrix}}\frac{r_{10;5}}{r_{5}}A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}BX_{2}A_{12,{1;D}}} \right\}}} \\ {= {\Pr\begin{Bmatrix} {\frac{r_{6;2}}{r_{6}}A_{6,{0;2}}*\left( {\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}*\frac{r_{8;4}}{r_{8}}A_{8,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}} \right)*\left( {\frac{r_{10;5}}{r_{5}}A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}} \right){BX}_{2}A_{12,{1;D}}} \\ {{+ \frac{r_{6;2}}{r_{6}}}A_{6,{0;2}}*\frac{r_{7,4}}{r_{7}}A_{7,{1;4}}\frac{r_{4;2}}{r_{4}}A_{4;2}*\left( {\frac{r_{8;5}}{r_{8}}A_{8,{1;5}}*A_{10,{1;5}}\frac{r_{5;2}}{r_{5}}A_{5;2}} \right){BX}_{2}A_{12,{1;D}}} \end{Bmatrix}}} \\ {= {\Pr\left\{ {\begin{pmatrix} \left( {\frac{1}{1}A_{6,{0;2}}*\left( {\frac{1}{1}A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}\frac{1}{1}A_{4;2}} \right)*\left( {\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;2}} \right)} \right. \\ {{+ \frac{1}{1}}A_{6,{0;2}}*\frac{1}{1}A_{7,{1;4}}\frac{1}{1}A_{4;2}*\left( {\frac{1}{2}A_{8,{1;5}}*\frac{1}{1}A_{10,{1;5}}\frac{1}{1}A_{5;2}} \right)} \end{pmatrix}{BX}_{2}A_{12,{1;D}}} \right\}}} \\ {= {\Pr\left\{ {A_{6,{0;2}}*\begin{pmatrix} {\left( {A_{7,{1;4}}*\frac{1}{2}A_{8,{1;4}}A_{4;2}} \right)*\left( {A_{10,{1;5}}A_{5;2}} \right)} \\ {{+ A_{7,{1;4}}}A_{4;2}*\left( {\frac{1}{2}A_{8,{1;5}}*A_{10,{1;5}}A_{5;2}} \right)} \end{pmatrix}A_{2;2}B_{2}A_{12,{1;D}}} \right\}}} \\ {= {a_{6,{0;2}}*\begin{pmatrix} {\left( {a_{7,{1;4}}*\frac{1}{2}a_{8,{1;4}}a_{4;2}} \right)*\left( {a_{10,{1;5}}a_{5;2}} \right)} \\ {{+ a_{7,{1;4}}}a_{4;2}*\left( {\frac{1}{2}a_{8,{1;5}}*a_{10,{1;5}}a_{5;2}} \right)} \end{pmatrix}a_{2;2}b_{2}a_{12,{1;D}}}} \\ {= {\left( {{\begin{matrix} 1 & {1 - 0.6 - 0.4} \end{matrix}1} - 0.3 - 0.7} \right)*\begin{pmatrix} \left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}*\frac{1}{2}\begin{pmatrix}  - & 0.8 \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.5 & 0.5 \end{pmatrix}} \right) \\ {*\left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.4 & 0.8 \end{pmatrix}} \right)} \\ {{+ \left( {\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.5 & 0.5 \end{pmatrix}} \right)}*} \\ \left( {\frac{1}{2}\begin{pmatrix}  - & 0.7 \end{pmatrix}*\begin{pmatrix}  - & 0.7 \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 0.4 & 0.8 \end{pmatrix}} \right) \end{pmatrix}\begin{pmatrix}  - & - & - \\  - & 1 & - \\  - & - & 1 \end{pmatrix}\begin{pmatrix}  - \\ 0.01 \\ 0.02 \end{pmatrix}0.1}} \\ {= 0} \end{matrix}$

According to the algorithms in [8], the weighting coefficients

${\xi_{i}(y)} = \frac{\zeta_{i}(y)}{\sum\limits_{j}{\zeta_{j}(y)}}$

of the sub-graphs are:

${{\xi_{1}(2)} = {\frac{\zeta_{1}(2)}{{\zeta_{1}(2)} + {\zeta_{2}(2)}} = {\frac{{0.0}09029}{{{0.0}09029} + 0} = 1}}}{{\xi_{2}(2)} = {\frac{\zeta_{2}(2)}{{\zeta_{1}(2)} + {\zeta_{2}(2)}} = {\frac{0}{{{0.0}09029} + 0} = 0}}}$

Since ζ₂(0)=0 or ξ₂(0)=0, FIG. 15 does not hold and should be eliminated. FIG. 14 is the only valid sub-graph, which means, BX_(1,1) is the only event in S_(H)(2). According to 10, the ranking procedure ends. The cause of abnormality of the current system is verified as BX_(1,1). 

1. The method to rank the state pending X-type variables with at least one CPU, according to the rank, part of or all of the state pending X-type variables' states which form E⁺(y) are detected sequentially or parallel, in order to find the real cause H_(kj) that is in S_(H)(y+1) and to rank the real H_(kj) as high as possible conditioned on E(y+1)=E⁺(y)E(y), detailed steps include: (1) determine the detectable state pending X-type variables whose index set is denoted as S_(X)(y) based on the simplified DUCG conditioned on E(y); (2) the ranking ends if S_(X)(y) contains only one element; (3) calculate the rank importance I_(i)(y) of X_(i); (4) rank X_(i) (i∈S_(X)(y)) according to I_(i)(y) and detect the states of X_(i) (i∈S_(X)(y)) in reference to the rank; (5) if the ranking is still needing, increase y to y+1, and repeat the above step (1)-(5) until the diagnosis is satisfied or no state pending X-type variable available.
 2. The method according to claim 1, wherein to determine the state pending X-type variables with at least one CPU, the detailed steps include: (1) Collect all possible H_(kj) based on the simplified DUCG conditioned on E(y), these H_(kj) form S_(H)(y); (2) For each H_(k) in S_(H)(y), search for the state pending X-type variable connected to H_(k) without state-known variables blocking them, where the indices of such X-type variables make up S_(X)(y).
 3. The method according to claim 1, wherein to determine the structure importance λ_(i)(y)>0 of X_(i) for calculating I_(i)(y) with at least one CPU, characterized in that: for the state pending X-type variables in the above 1(1), count the number of its connected different H_(k) (k∈S_(iK)) in S_(H)(y) determined in the above 2, the number is written as m_(i)(y), calculate λ_(i)(y) based on m_(i)(y) according to a method that features at that the bigger m_(i)(y) is, the smaller λ_(i)(y) is, such method includes but not limits to λ_(i)(y)=1/(m_(i)(y))^(n) (n=1, 2, . . . ).
 4. The method according to claim 1, wherein assign a value to the danger importance ω_(kj)>0 of H_(kj) for calculating I_(i)(y) with at least one CPU, characterized in that: for each possible cause event H_(kj) in S_(H)(y), score all of the abnormal states according to their degree of concern, the score is called concern importance which can be written as ω_(k), 1≥ω_(k)>0. The greater the concern, the bigger ω_(k) is. The value of ω_(k) can be assigned when constructing DUCG or be assigned according to the concrete situation given H_(k) when S_(H)(y) is known.
 5. The method according to claim 1, wherein to calculate the probability importance ρ_(i)(y) for calculating I_(i)(y) with at least one CPU, characterized in that: calculate the average variation of the conditional probabilities of H_(k) in S_(H)(y) between the conditions E(y) and X_(ig)E(y) over all abnormal states of all H_(k) connected with X_(i) without state-known variables blocking them, where g∈S_(iG)(y), S_(iG)(y) is the index set of possible states of X_(i), and X_(i) is non-cause-specific state pending X-type variables, based on the simplified DUCG conditioned on E(y), featuring at that the greater the average variation is, the bigger ρ_(i)(y) is, such as but are not limited to: ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}{Pr\left\{ X_{ig} \middle| {E(y)} \right\}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}}$ Or ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}\left( {Pr\left\{ X_{ig} \middle| {E(y)} \right\}\left( {{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}} \right)} \right)^{2}}}}}}$ Or ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}{{{\Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \middle| {E(y)} \right\}}}}}}}}}$ Or ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}\sqrt{\sum\limits_{j \in {{S_{kJ}{(y)}}g} \in}{\sum\limits_{S_{iG}{(y)}}\left( {Pr\left\{ {{H_{kj}\left. {X_{ig}{E(y)}} \right\}} - {Pr\left\{ H_{kj} \right.{E(y)}}} \right\}} \right)^{2}}}}}}$ Or ${\rho_{i}(y)} = {\frac{1}{m_{i}(y)}{\sum\limits_{k \in {S_{iK}{(y)}}}{\omega_{k}{{{\sum\limits_{g \in {S_{iG}{(y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}} - {\frac{1}{m_{i}}{\sum\limits_{k \in {{S_{iK}{(y)}}j} \in {S_{kJ}{(y)}}}{\sum\limits_{g \in {S_{iG}{\langle y)}}}{Pr\left\{ H_{kj} \middle| {X_{ig}{E(y)}} \right\}}}}}}}}}}$ E(y) and ω_(k) can be ignored, equivalent to being removed separately or together, that is to let E (y)=the complete set and ω_(k)=1.
 6. The method according to claim 1 for calculating I(y) with at least one CPU, wherein to determine the cost importance, characterized in that: comprehensively assign a cost score for the state pending X_(i) (i∈S_(X)(y)), or calculate the cost score as the sum of the weighted scores assigned for the difficulty of doing the detection (indexed by j=1), waiting time (indexed by j=2), price (indexed by j=3) and damage to target system (indexed by j=4) respectively, the weights σ_(ij) can be given when constructing DUCG or be given in an individual application, the bigger the cost score is, the smaller the cost importance β_(i) (1≥β_(i)>0) is, which including but not limited to: when the highest cost score is 100, β_(i)= 1/100=0.01, when the lowest cost score is 1, β_(i)=1/1=1, and the remnant values are between the highest and the lowest case, β_(i) can be assigned when constructing DUCG or be given online for the state pending X-type variables included in the simplified DUCG conditioned on E(y).
 7. The method according to claim 5 to determine the probability importance ρ_(i)(y) with at least one CPU, characterized in that: based on the simplified DUCG conditioned on E(y), for each cause-specific state pending X-type variable connected to H_(kj) (H_(kj)∈S_(k)(y)), λ_(i)(y) and ρ_(i)(y) are not calculated according to the above 3 and 5, which means S_(Xs)(y) is obtained by subtracting S_(s)(y) from S_(X)(y), but are calculated by selecting the maximum of i∈S_(s)(y), which includes but is not limited to λ_(i)(y)≥1 and ${{\rho_{i}(v)} \geq {\max\limits_{l \in {S_{Xs}{(y)}}}\left\{ {\rho_{l}(y)} \right\}}},$ that means X_(i) in the above 5 is limited to i∈S_(Xs)(y).
 8. The method according to claim 1 to comprehensively calculate I_(i)(y) with at least one CPU, characterized in that: the bigger λ_(i)(y) or ρ_(i)(y) or in claims 2-7 is, the bigger I_(i)(y) is, the detailed calculation formulas include but are not limited to: ${I_{i}(y)} = \frac{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}\beta_{i}{\rho_{i}(y)}}}$ Or I_(i)(y) = λ_(i)(y)β_(i)ρ_(i)(y) Or ${I_{i}(y)} = \frac{{\lambda_{i}(y)}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{{\lambda_{i}(y)}{\rho_{i}(y)}}}$ Or I_(i)(y) = λ_(i)(y)ρ_(i)(y) Or ${I_{i}(y)} = \frac{\beta_{i}{\rho_{i}(y)}}{\sum_{i \in {S_{X}{(y)}}}{\beta_{i}{\rho_{i}(y)}}}$ Or I_(i)(y) = β_(i)ρ_(i)(y) Or ${I_{i}(y)} = \frac{\rho_{i}(y)}{\sum_{i \in {S_{X}{(y)}}}{\rho_{i}(y)}}$ Or I_(i)(y) = ρ_(i)(y) Or I_(i)(y) = w₁λ_(i)(y) + w₂ρ_(i)(y) + w₃β_(i) Or ${I_{i}(y)} = \frac{{w_{1}{\lambda_{i}(y)}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}{{\sum_{i \in {S_{X}{(y)}}}{w_{1}{\lambda_{i}(y)}}} + {w_{2}{\rho_{i}(y)}} + {w_{3}\beta_{i}}}$ where w₁, w₂, and w₃ are three weights, and w_(i)≥0 (i=1, 2, 3), and w_(i)=0 means this item is not considered, wherein the value of w₁, w₂, and w₃ can be assigned when constructing DUCG or be assigned according to the individual application situation.
 9. The method according to claim 1 to rank the state pending X-type variables X_(i) (i∈S_(X)(y)) according to I_(i)(y) with at least one CPU, characterized in that: in the ranking, when the top-ranking X-type variable is the only ancestor or descendant variable of the ranking lower X-type variable, the ranking lower X-type variable is eliminated from the current rank, also the X whose I_(i)(y)=0 is eliminated from the rank.
 10. The method according to claim 1 to determine whether the ranking ends or not with at least one CPU, characterized in that: the ranking ends if there is only one hypothesis event in S_(H)(y), or if all the state pending X-type variables are state-known, or there is only one element in S_(X)(y). 